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Smooth representations of epi-Lipschitzian subsets of \(\mathbb{R}^n\). (Représentations lisses de sous-ensembles épi-lipschitziens de \(\mathbb{R}^n\).) (French. Abridged English version) Zbl 0893.49012
Summary: Closed epi-Lipschitzian subsets \(M\) of \(\mathbb{R}^n\) are characterized as sets defined by a Lipschitzian inequality constraint \(\{x\in \mathbb{R}^n\mid f_M(x)\leq 0\}\) for some function \(f_M:\mathbb{R}^n\to \mathbb{R}\) which is Lipschitzian on \(\mathbb{R}^n\), “smooth” \((C^\infty)\) on the complementary of \(\partial M\) (the boundary of \(M\)), which satisfies, for every \(x\in\partial M\), both the “nondegeneracy” condition: \(0\not\in\partial f_M(x)\) (Clarke’s subdifferential), and the “normal representation” condition: that \(N_M(x)\) (Clarke’s normal cone) is the cone spanned by \(\partial f_M(x)\). This geometrical characterization is also equivalent to a more analytic formulation only by using the function \(\Delta_M= d_M- d_{\mathbb{R}^n\backslash M}\) (where \(d_M\) is the distance function to \(M\)).
Applications of this result are given here and in [the authors, “Smooth representations of epi-Lipschitzian subsets of \(\mathbb{R}^n\)”, Nonlinear Anal., Theory Methods Appl. (to appear); “Smooth normal approximation of epi-Lipschitzian subsets of \(\mathbb{R}^n\)”, C. R. Acad. Sci., Paris, Sér. I, Math. (to appear); “Existence of (generalized) equilibria on an epi-Lipschitz domain: a necessary and sufficient condition”, Cahier Eco-Maths No. 95-55, Université de Paris 1 (1995), per bibl.] to (i) the smooth (normal) approximation of epi-Lipschitzian sets; (ii) Green’s formula in the nonsmooth case; and (iii) the study of “variational inequalities” (or “generalized equations”) in the nonconvex and nonsmooth case.

MSC:
49J52 Nonsmooth analysis
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